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Abstract

The theory of second harmonic generation (SHG) in three-dimensional structures consisting of arbitrary distributions of metallic spheres made of centrosymmetric materials is developed by means of multiple scattering of electromagnetic multipole fields. The electromagnetic field at both the fundamental frequency and second harmonic, as well as the scattering cross section, are calculated in a series of particular cases such as a single metallic sphere, two metallic spheres, chains of metallic spheres, and other distributions of the metallic spheres. It is shown that the linear and nonlinear optical response of all ensembles of metallic spheres is strongly influenced by the excitation of localized surface plasmon-polariton resonances. The physical origin for such a phenomenon has also been analyzed.

Figures (9)

Geometry of the scattering problem for an ensemble of N spheres consisting of centrosymmetric materials. Here θn, ϕn and rn are the coordinate of the nth sphere in the spherical coordinate system, Ω=(θ,ϕ) represents the solid angle of an arbitrary point P. The x, y and z represent three axis directions in corresponding Cartesian coordinate system.

(Color online) The spatial profile of the amplitude of the electric field, calculated at ℏω=5.0eV (panels A and B) and ℏω=2.7eV (panels C and D). The radius of the sphere is a = 10 nm. A and C correspond to the FF; B and D to the SH.

The spectra of the scattering cross sections for the metallic two-sphere system. The radii of two spheres are taken as a = 10 nm. (I) and (II) correspond to the separation distance d = 22nm, (III) and (IV) to d = 35nm. Solid line and dashed line correspond to the case at the incident direction of the wave perpendicular or parallel to the longitudinal axis of the system, respectively. (I) and (III): the FF; (II)and (IV): the SH.

(Color online) The spatial profile of the amplitude of the electric field for the two-sphere system, calculated at ℏω=4.1eV for the normal incidence with d = 22nm (panels A and B) and d = 35nm (panels C and D); calculated at ℏω=6.54eV for the parallel incidence with d = 22nm (panels E and F) and d = 35nm (panels G and H). The radii of the spheres are a = 10 nm. A, C, E and G correspond to the FF; B, D, F and H to the SH. Here A, B, C, D, E, F, G and H correspond to the points marked in Fig. 3.

(Color online) The top two panels show the spectra of the scattering cross section corresponding to a chain of N = 12 metallic spheres under the parallel incidence. The radius of the sphere is a = 10 nm and the separation distance is d = 22nm. (I): the FF; (II): the SH. The spatial profile of the amplitude of the electric field, calculated at ℏω=5.3eV ((A) and (B)) and ℏω=2.82eV ((C) and (D)), is presented in the bottom panels. The panels A and C correspond to the FF, whereas the panels B and D correspond to the SH. Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

(Color online) The same as in Fig. 6, but for the normal incidence. The spatial profile of the amplitude of the electric field, calculated at ℏω=5.4eV ((A) and (B)) and ℏω=2.59eV ((C) and (D)). Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

(I) The 3D sample and cross sections. (II) and (III) The spectra of the scattering cross sections corresponding to the cluster as shown in (I) under the incidence of the plane wave from the left side of the sample with a = 10 nm and d = 22nm. (II): the FF; (III): the SH.

The spatial profiles of the amplitude of the electric field, calculated at ω=2.21eV (corresponding to the points A and B in Fig. 7 (II) and (III)). (a) and (b) correspond to the FF and the SH fields at the cross-section as shown by shadow in Fig. 7 (I), respectively, whereas (c) and (d) correspond to the FF and the SH fields at the cross-section as shown by dot-dashed lines in Fig. 7 (I). The other parameters are identical with those in Fig. 7.